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Use Strategy C25 from Unit C4 to find the equation of the quadric x² + 3y² +52²-8xy-8yz + 3y +3z-6=0 in standard form. Which of the six types of non-degenerate quadric is this quadric?

Use Strategy C25 from Unit C4 to find the equation of the quadric x² + 3y² +52²-8xy-8yz + 3y +3z-6=0 in standard form. Which of the six types of non-degenerate quadric is this quadric?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Use Strategy C25 from Unit C4 to find the equation of the quadric x² + 3y² +52²-8xy-8yz + 3y +3z-6=0 in standard form. Which of the six types of non-degenerate quadric is this quadric?
Transcribed Image Text:Use Strategy C25 from Unit C4 to find the equation of the quadric x² + 3y² +52²-8xy-8yz + 3y +3z-6=0 in standard form. Which of the six types of non-degenerate quadric is this quadric?
Strategy C25 To write the quadric with equation Ax² + By² + Cz²+Fxy + Gyz + Hxz+Jx+Ky+Lz + M = 0 in standard form, do the following. 1. Introduce matrices: write down the matrices AF H }F BG HG C A = and J = 2. Align the axes: • orthogonally diagonalise A to get A₁0 0 0 20 0 0 A3/ PT AP = J K L find (f g h) = JTP, and write the quadric in the form A₁ (x¹)² + A₂(y)² + A3(2')² + fx' + gy' + hz' + M = 0. 3. Translate the origin: complete the squares • make a substitution to change to the coordinate system (x", y", z").
Transcribed Image Text:Strategy C25 To write the quadric with equation Ax² + By² + Cz²+Fxy + Gyz + Hxz+Jx+Ky+Lz + M = 0 in standard form, do the following. 1. Introduce matrices: write down the matrices AF H }F BG HG C A = and J = 2. Align the axes: • orthogonally diagonalise A to get A₁0 0 0 20 0 0 A3/ PT AP = J K L find (f g h) = JTP, and write the quadric in the form A₁ (x¹)² + A₂(y)² + A3(2')² + fx' + gy' + hz' + M = 0. 3. Translate the origin: complete the squares • make a substitution to change to the coordinate system (x", y", z").
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